On a Relation Between Randić Index and Algebraic Connectivity

A conjecture of AutoGraphiX on the relation between the Randić index R and the algebraic connectivity a of a connected graph G is: R a ≤ ( n− 3 + 2√2 2 ) / ( 2(1− cos π n ) ) with equality if and only if G is Pn, which was proposed by Aouchiche et al. [M. Aouchiche, P. Hansen and M. Zheng, Variable neighborhood search for extremal graphs 19: further conjectures and results about the Randić index, Match Commun. Math. Comput. Chem. 58(2007), 83–102.]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity κ′(G) ≥ 2, and if κ′(G) = 1, the conjecture holds for sufficiently large n. The conjecture also holds for all connected graphs with diameter D ≤ 2(n−3+2 √ 2) π2 or minimum degree δ ≥ n2 . We also prove R · a ≥ 8 √ n−1 nD2 and R · a ≥ nδ(2δ−n+2) 2(n−1) , and then R · a is minimum for the path if D ≤ (n− 1)1/4 or δ ≥ n2 . ∗Corresponding author.